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A local energy estimate for 2-dimensional Dirichlet wave equations

Kellan Hepditch and Jason Metcalfe

Vol. 16 (2023), No. 3, 483–492

We examine a variant of the integrated local energy estimate for (1+2)-dimensional Dirichlet wave equations exterior to star-shaped obstacles. The classical bound on the solution, rather than the derivative, is not typically available in two spatial dimensions. Using an argument inspired by the rp-weighted method of Dafermos and Rodnianski and taking advantage of the Dirichlet boundary conditions allow for the recovery of such a term when the initial energy is appropriately weighted.

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wave equation, local energy estimate, exterior domain
Mathematical Subject Classification
Primary: 35L71, 35L05
Received: 7 February 2022
Revised: 15 June 2022
Accepted: 11 September 2022
Published: 10 August 2023

Communicated by Suzanne Lenhart
Kellan Hepditch
Cedar Ridge High School
Hillsborough, NC
United States
Jason Metcalfe
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States